Geometrical insight into non-smooth bifurcations of a soft impact oscillator

In this paper, the concept of discontinuity geometry of rigid impact oscillators is extended to soft impact oscillators and the mechanisms of grazing bifurcations of an impact oscillator with one-sided elastic constraint are studied by the geometric and dynamical systems methods. The existence conditions of periodic solutions are given by the discontinuity geometry objects and then are used to derive the discontinuity curves. Several bifurcation scenarios near grazing bifurcation are studied to examine the evolution of the system dynamics from a geometrical point of view. The geometrical conditions are obtained for the existence of periodic orbits with one impact, grazing and saddle-node bifurcations. Some geometrical insight is gained into a question whether there is a discontinuous jump or a continuous transition from a non-impacting period-1 to an impacting period-1 attractor at a grazing bifurcation.